Chocolate Malt Sandwich Cookies

For the cookies:

• 2 cups plus 2 tablespoons all-purpose flour
• 1/2 cup unsweetened Dutch-process cocoa powder
• 1/4 cup plain malted-milk powder
• 1 teaspoon baking soda
• 1/2 teaspoon coarse salt
• 1 cup (2 sticks) unsalted butter, softened
• 1 3/4 cup sugar
• 1 large egg
• 2 teaspoons vanilla extract
• 1/4 creme fraiche (or sour cream)

For the filling:

• 10 ounces semi-sweet chocolate, coarsely chopped
• 4 tablespoons unsalted butter, cut into small pieces
• 1 cup malted-milk powder
• 3 ounces cream cheese, room temperature
• 1/4 cup plus 2 tablespoons half-and-half
• 1 teaspoon pure vanilla extract